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For $\alpha\in (0,2)$ we study the null controllability of the parabolicoperator $$Pu= u_t - (\vert x \vert ^\alpha u_x)_x\qquad (1<x<1),$$ whichdegenerates at the interior point $x=0$, for locally distributed controlsacting only one side of the origin (that is, on some interval $(a,b)$ with$0<a<b<1$). Our main results guarantees that $P$ is null controllable if andonly if it is weakly degenerate, that is, $\alpha \in (0,1)$. So, in order tosteer the system to zero, one needs controls to act on both sides of the pointof degeneracy in the strongly degenerate case $\alpha\in [1,2)$.  Our approach is based on spectral analysis and the moment method. Indeed, wecompletely describe the eigenvalues and eigenfunctions of the associatedstationary operator in terms of Bessel functions and their zeroes for bothweakly and strongly degenerate problems. Hence, we obtain lower $L^2$ boundsfor the eigenfunctions on the control region in the case $\alpha \in [0,1)$ anddeduce the lack of observability in the case of $\alpha \in [1,2)$. We alsoprovide numerical evidence to illustrate our theoretical results.

Null controllability of parabolic equations with interior degeneracy and one-sided control